Exercise 3. Is the following argument valid or invalid? Justify your answer.

p ? q

p ? r

q ? r

? r

Exercise 4. Prove that The checkbook was on the breakfast, given the following premises (axioms):

(i) If my checkbook is on my office table, then I paid my phone bill. (ii) I was looking at the phone bill for payment at breakfast or I was looking at the phone bill for payment in my office. (iii) If I was looking at the phone bill at breakfast, then the checkbook is on the breakfast table. (iv) I didnt pay my phone bill. (v) If I was looking at the phone bill in my office, then the checkbook in on my office table. In order for you to discuss this exercise easily with other classmates, lets use following:

p: My checkbook in on my office table.

q: I paid my phone bill.

r: I was looking at the phone bill for payment at breakfast.

s: I was looking at the phone bill for payment for payment in my office.

t: The checkbook is on the breakfast table.

Exercise 5. Find a counterexample to show that the following statements are false.

a): ?a, b ? R, ? a + b = ? a + ? b

b): ?a, b ? R, ? ab = ? a ? b

Exercise 6. In this exercise the predicate P(x, y) denotes x + y = 1. What is the truth values of the following statements where x and y are integers.

a): ?x?y P(x, y)

b): ?x?y P(x, y)

c): ?x?yP(x, y)

Exercise 7. Is the the argument below valid? Justify your answer.

?x(Q(x) ? F(x))

?x(Z(x) ? Q(x))

? ?x(Z(x) ? F(x))

Exercise 8. Given that x ? Z, prove that x 2 ? x is an even integer.

Exercise 9. Let x, y ? Q ? . Prove that 5x + 2y 3y ? Q

Exercise 10. Prove or disprove: If x /? Q then x 2 ?/ Q

Exercise 11. Prove by contradiction that if r ? Q ? and s /? Q then r s /? Q

Exercise 12. By direct proof, show that if a + b + ab = 0 then a = b = ?2 where a, b ? Z ?

Exercise 13. a) Let m, n ? Z, and suppose that n 2 + 1 = 2m. Prove that m is the sum of the squares of two integers. b)Write 9, 561, 565 as the sum of two squares. Note that 43732 + 1 = 2 9, 561, 565

Exercise 14. a) Give a direct proof of the fact that a 2 ? 5a + 6 is even for any integer a.

b) Suppose a, b ? Z and a 2 ? 5b is even. Prove that b 2 ? 5a is even